Theorem ltrnmw

Description: Property of lattice translation value. Remark below Lemma B in Crawley p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	ltrnmw.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrnmw.m	⊢ ∧ = ( meet ‘ 𝐾 )
	ltrnmw.z	⊢ 0 = ( 0. ‘ 𝐾 )
	ltrnmw.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrnmw.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnmw.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrnmw	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∧ 𝑊 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		ltrnmw.l	⊢ ≤ = ( le ‘ 𝐾 )
2		ltrnmw.m	⊢ ∧ = ( meet ‘ 𝐾 )
3		ltrnmw.z	⊢ 0 = ( 0. ‘ 𝐾 )
4		ltrnmw.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		ltrnmw.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		ltrnmw.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8	1 4 5 6	ltrnel	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
9	1 2 3 4 5	lhpmat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∧ 𝑊 ) = 0 )
10	7 8 9	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∧ 𝑊 ) = 0 )